For requesting, clarifying, and comparing definitions of mathematical terms.

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2
votes
1answer
46 views

Mathematical definition of the word “generic” as in “generic” singularity or “generic” map?

I've been trying to work out what generic means but I'm not making much progress. You can find an example of the usage of the word generic for example here: "School on Generic Singularities in ...
2
votes
3answers
63 views

Can limits be defined in a more algebraic way, instead of using the completely analytic $\delta$-$\epsilon$ definition?

Let $(X,d_X), (Y,d_Y)$ be metric spaces. Let $f:E\subseteq X\to Y$ and $a\in X$. We say that $\lim_{x\to a}f(x)=L$ if and only if for every $\epsilon >0$ there exists a $\delta>0$ such that ...
1
vote
2answers
54 views

Does defining a type of mathematical object require defining a binary relation of “equality”?

I'm trying to determine whether defining a type of mathematical object requires us to know what we mean by another object being "equal" to it. For example, when we define a type of object like set, ...
1
vote
0answers
30 views

Étale morphisms definition?

Working over a commutative ring $R$, let $D= \left\{ d\in R : d^2 =0 \right\}$ A formally étale morphism $f:M\rightarrow N$ is one for which the square below is a pullback for every point $d:\mathbf{1}...
0
votes
1answer
40 views

What are “members” of an event in probability theory?

My homework asks a question along the lines of "Suppose a fair six-sided die is rolled and a fair coin is flipped. List the members of the event 'the die lands on an even number.'" I can only guess ...
1
vote
1answer
23 views

Definition of a monotone class

The way I learned it, a monotone class on a set $X \neq \varnothing$ is a collection of subsets of $X$ that is closed under monotone countable unions and intersections. According to this, $X$ is not ...
2
votes
1answer
29 views

What is the closure of $A = \{x| 1<\|x\|<3\} \cup \{(0,0)\}$ and why am I wrong?

Given $A = \{x| 1<\|x\|<3\} \cup \{(0,0)\}$ Find $\bar A$ My hunch is $\overline A = \{x| 1 \leq x \leq 3\} \cup \{0,0\}$, but my friend says I am wrong, the closure of $A$ must be $...
0
votes
1answer
30 views

Nothing on the web; What is a Ruffini Radical

Surprisingly, it's not clearly defined online. The first thing that comes up is Abel-Ruffini theorem, which only refers to "radicals" and not RUFFINI radicals. Ian Stewart's book has it appear out of ...
1
vote
1answer
33 views

finite presentations of algebraic structures

I've read an algebraic variety $A$ is finitely presented if there's a coequalizer diagram $$F(m)\rightrightarrows F(n)\twoheadrightarrow A$$ where $F(n)$ is the free model on $n$ generators. I'm ...
6
votes
0answers
38 views

Why do isotropic spaces deserve their name?

Wiki defines a quadratic form to be isotropic if it evaluates to zero at some vector. What does this have to do with isotropy in physics i.e uniformity in all directions? From my experience so far, ...
0
votes
1answer
52 views

Why is $\Delta x \approx \left(\dfrac{\partial x}{\partial u}\right)_0\Delta u$?

I'm trying to understand the proof for variable substitution of multivariable integrals here, but I'm not really sure what is meant by $\Delta x \approx \left(\dfrac{\partial x}{\partial u}\right)_0\...
0
votes
1answer
26 views

Mathematical definition of congruent sets

I cant really find a proper definition for this. If two sets are congruent, then what does that mean. I heard that it can be defined in terms of isometries... This is with respect to the banach ...
3
votes
0answers
38 views

A Cartesian coordinate system is a mathematical or physical thing?

I'm convinced that if I ask what of the coordinates systems in the figure is a Cartesian system almost all say that it is the system $O_1$. This answer comes immediately from our habit and ...
3
votes
0answers
47 views

С* algebras and projective limits

Let $A_i$ be a family of $C^*$-algebras, and let $\varphi_{ij} = A_i \leftarrow A_{j}$ be $*$-morphisms which form some projective system. How can we define a $C^*$-(pre)norm on a projective algebraic ...
1
vote
2answers
52 views

Is a function differentiable in $x$ if $\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}=\infty$?

All the definitions of differentiability I found (Wolfram Mathworld for instance) only require this limit to exist, but say nothing about the domain in which that has to happen. So what if that ...
1
vote
0answers
42 views

Is this how we define “limit in the distributional sense”?

Consider $\mathcal{D}(\mathbb{R})$ the space of test functions and $\mathcal{D}'(\mathbb{R})$ the space of distributions, in $\mathbb{R}$, i.e., continuous linear functionals over $\mathcal{D}(\mathbb{...
2
votes
0answers
38 views

Does the metric in the Theory of Relativity actually satisfy the definition of a metric?

Allow me to give a brief introduction to the topic, which has to do with physics; my question will still be a mathematical one. I think my question is aimed at people with a background both in physics ...
2
votes
2answers
278 views

Are these two definitions of differentiability equal?

A function $f$ is differentiable in $x$ iff the limit $~\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}~~\text{exists}$ ("normal" definition) $|f(x+h)-f(x)|<C|h|~$ holds for small $h$ with $...
2
votes
0answers
34 views

Is the generalized mandelbrot set a fractal in the $d$ dimension?

The $d$-mandelbrot set is defined as the set of $c$ such that the iterations of $$z \mapsto z^d + c$$ starting with $z=0$ is bounded in absolute value. Here is a picture of the mandelbrot sets from $...
2
votes
1answer
31 views

Quick clarification on the definition of vector field

I am having a class on differential geometry and another on ODEs In the ODE class, if we were given something of the type $$\dot x = x^2$$ The professor refers to $x^2$ as the vector field. In ...
5
votes
2answers
78 views

Probability: mathematically what does it mean to say “let $X$ be a random variable WITH a cdf/pdf”

I don't quite understand what people mean by let "$X$ be a random variable WITH a cdf/pdf". For example, there is a question that says: "Let X be a random variable with the 3-parameter Weibull pdf and ...
0
votes
1answer
40 views

Why do we need $\sigma$-algebra in the definition of a measurable space?

A measurable space is a triple $(M, \Sigma, \mu)$. I am confused by one thing, to my knowledge almost everything you think of is measurable. So why do we need $\Sigma$? Why is that a measurable set ...
1
vote
1answer
43 views

What is the group $O(2)$?

I have learnt about the group $$ SO(2) = \left\{\pmatrix{\cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta)} : \theta \in \mathbb{R} \right\}. $$ This group has do with rotations. ...
0
votes
1answer
17 views

What is “analytic vector for closed operator”?

I need the defenition of "analytic vector of closed operator that acts on Hilbert space". I cant find it in google and in my textbooks (Khelemsky "Lectures And Exercises on Functional Analysis"), I ...
0
votes
1answer
38 views

Matching the definition of hom-functor with how these are used when defining adjuncts

I have a problem matching the definition of a hom-functor (from nlab) with how this concept it used in the definition of adjunction (from nlab): The hom-functor is defined on $C^{\text{op}}\times C$, ...
1
vote
1answer
55 views

Defining adjoint functors: What does “natural bijection” mean?

Take the following definition of adjunction from the nlab This definition can be found in numerous other places. My brain parses this definition perfectly up till the point where it says that for ...
0
votes
2answers
48 views

Is $f(x) = 2x+1$ injective? Is it surjective? [closed]

How would I answer this? I know what it means to be surjective and injective. Is the function $f(x)=2x+1$ injective? Is it surjective? Give reasons for your answers. I assume they are both because ...
0
votes
1answer
25 views

Meaning of denseness in a $L^p$ spaces?

I am currently studyind Density theorems in $L^p$ - spaces. In that, I have encountered a theorem which goes like this - The space of integrable simple functions is dense in $L^p $(E, $\mathcal{A}$ ,...
1
vote
2answers
33 views

Problem in understanding definition of absolutely continuous?

Suppose $(E, \mathcal{A})$ is a measurable space. Let $\mu$ and $\gamma$ be two distinct measures of this space. Now we say that $\gamma$ is absolutely continuous with respect to $\mu$ if for every $A ...
1
vote
0answers
70 views

What is the name of the property $x^m=x$ when $x$ is in a ring?

I have been doing problems in Atiyah & MacDonald's Introduction to Commutative Algebra, and in problem 1.6 it asks to assume the existence of an idempotent element in an ideal whenever the ideal ...
5
votes
4answers
419 views

Definition of the total derivative.

I am trying to understand the following definiton. $f:\mathbb{R}^n \rightarrow \mathbb{R}^m$ . The total derivative of $f$ in point $a$ is the unique linear map $Df|_a$ such that $$\lim_{h \...
1
vote
1answer
32 views

Definition request: explicit definition of covering compactness in terms of set notation

Part of my confusion with covering compactness stems from the fact that it is a definition given almost completely in a high level manner (in English no less). When I look at: A set $A \subset (...
2
votes
4answers
68 views

What is the exact definition of a metric?

In some book I found, a metric on a non-empty set $X$ defined as a map $$X\times X\to \Bbb R^{+}$$ and some other place as $$X\times X\to \Bbb R$$ So, is a metric a real valued function or a non-...
0
votes
0answers
22 views

Formal definition of a face of a polyhedron

Given an $n$-dimensional convex polyhedron, an $(n-1)$-dimensional face of it can be defined as an intersection of the polyhedron with a supporting hyperplane. What is the formal definition in the ...
1
vote
1answer
23 views

If $f:\mathbb{R}^2\to X$, how do we call $f_{x_0}(y) =f(x_0,y)$ as a function of one variable?

For obvious reasons coming from probability theory I have been calling the function $f_{x_0}(y)$ ($x_0$ fixed) "marginal function". However, reviewing some literature I've noticed that the word "...
1
vote
0answers
57 views

Ignoring the lack of rigor, is this a fair argument to make when considering if 0^0 should be equivalent to 1? [closed]

The Professor of Mathematics argued that 0^0 is undefined because the limits $0^x$ and $x^0$ as x approaches 0 don't agree. That seemed logical to me, but then Scott pointed out in the comments that ...
4
votes
4answers
164 views

Iff Interpretation

I understand that (1) "$A$ if and only if $B$" ($A\iff B)$ means that (2) "$A$ implies $B$ and $B$ implies $A$" $(A\implies B)\land (B\implies A)$. The phrase "$A$ if and only if $B$" sounds as ...
-1
votes
2answers
62 views

What does “closed” mean in Heine Borel for $C^0$?

Heine Borel for $C^0$: A set $\mathcal{E} \subseteq C^0([a,b], \mathbb{R})$ is compact if it is closed, bounded and equicontinuous. I don't really understand what closed mean in the definition....
7
votes
4answers
987 views

What is wrong with this argument that closed interval [0, 1] is not compact?

I am a student majoring engineering. I am studying real analysis with textbook 'Measure and Integral' by Wheeden and Zygmund. This book defined compact like the following: $E$ is compact if ...
4
votes
0answers
87 views

Why is cos at $\pi/2$ not undefined?

If the $\cos$ function is based off of the ratio of the adjacent side of Euclidean, right triangle, with fixed hypotenuse length (such as the unit circle), then how does this correspond to a defined ...
1
vote
1answer
128 views

How does one “join” two graphs in graph theory?

I am asked to find the join of two graphs in graph theory. But I cannot find the exact definition! I know that in lattice theory, we join every vertex of a graph to every vertex of another graph to ...
1
vote
2answers
50 views

$\nabla ^2 G$ meaning

If $G$ is a function with three components. $G$ is actually a Green's function in my case, like $G( \textbf{x} , \xi)$ with $\textbf{x} = (x,y)$ and $\xi = (\xi _x, \xi _y)$ so I am guessing it is ...
1
vote
1answer
23 views

Series representation of a differential form

I have a problem understanding the general series representation of a p-form. For 1-form things are pretty clear to me: For $h = (h_1, \dots, h_n)^T \in \mathbb{R}^n $ and $ h = \sum\limits_{i = 1}^{...
2
votes
3answers
102 views

Understanding iff [duplicate]

I'm having difficulty understanding why it is appropriate to use if and only if, something I thought I had a firm grasp on. From Lara Alcock's book, How to Study as a Mathematics Major: ...
1
vote
1answer
292 views

A proof that $|x+yi|=\sqrt{x^2+y^2}$, based on the given the conditions

If we attempt to define $|x+yi|$ by following conditions: $|x|=|xi|=x\operatorname{sgn}(x)$ (implicitly meaning the result will always be $\ge 0$) $|xz|=|x||z|$ $|z^x|=|z|^x$ for $x \in \mathbb{R}...
0
votes
1answer
42 views

I know what a Complex Plane is, but what is a complex $k$-plane?

This may be somewhat related to physics, but I saw in a non-English paper (which I googled "complex $k$-plane" for some constant real $k$) that mentioned a complex $k$-plane. $k$ in its context was ...
2
votes
1answer
71 views

What's the difference between a Nash, Correlated, and Extreme equilibrium?

As the title states, what's the difference? As I understand it: The Nash Equilbirum (NE) is a solution concept in non-cooperative games where no player has incentive to unilaterally deviate from a ...
21
votes
7answers
2k views

Proof that the empty set is a relation

In the book Naive Set Theory, Halmos mentions that the "The least exciting relation is the empty one." and proves that the empty set is a set of ordered pairs because there is no element of the empty ...
0
votes
1answer
24 views

Simpler way to define convex functions

I find the standard way of defining convex and concave functions slightly tricky. For me, it is the introduction of the new variable $0<λ<1$. However, I understand it intuitively. I was ...
0
votes
2answers
101 views

In definition of a category , what is the meaning of 'consists of'

A category $\mathsf C$ consists of the following three mathematical entities: A class $\operatorname{ob}(\mathsf{C})$, whose elements are called objects; A class $\hom(\mathsf{C})$, whose ...